Let be an arbitrary point on the graph of such that as shown in the figure. Verify each

Chapter 3, Problem 76

(choose chapter or problem)

Investigation Let \(P\left(x_{0}, y_{0}\right)\) be an arbitrary point on the graph of f such that \(f^{\prime}\left(x_{0}\right) \neq 0\), as shown in the figure. Verify each statement.

(a) The x-intercept of the tangent line is \(\left(x_{0}-\frac{f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)}, 0\right)\).

(b) The y-intercept of the tangent line is \(\left(0, f\left(x_{0}\right)-x_{0} f^{\prime}\left(x_{0}\right)\right)\).

(c) The x-intercept of the normal line is \(\left(x_{0}+\int\left(x_{0}\right) \int^{\prime}\left(x_{0}\right), 0\right)\).

(d) The y-intercept of the normal line is \(\left(0, y_{0}+\frac{x_{0}}{f^{\prime}\left(x_{0}\right)}\right)\).

(e) \(|B C|=\left|\frac{f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)}\right|\)

(f) \(|P C|=\left|\frac{f\left(x_{0}\right) \sqrt{1+\left[f^{\prime}\left(x_{0}\right)\right]^{2}}}{f^{\prime}\left(x_{0}\right)}\right|\)

(g) \(|A B|=\left|f\left(x_{0}\right) f^{\prime}\left(x_{0}\right)\right|\)

(h) \(|A P|=\left|f\left(x_{0}\right)\right| \sqrt{1+\left[f^{\prime}\left(x_{0}\right)\right]^{2}}\)

Text Transcription:

P(x_0),y_0)

f’(x_0) neq 0

(x_0-f(x_0)/f’(x_0), 0)

(0, f(x_0)-x_0 f’(x_0))

(x_0 + Int(x_0)Int’(x_0),0)

(0, y_0+x_0/f’(x_0))

|BC|=|f(x_0)/f’(x_0)|

|PC|=|f(x_0)sqrt 1 + [f’(x_0)]^2/f’(x_0)|

|AB|=|f(x_0)f’(x_0)|

|AP|=|f(x_0)|sqrt 1 + [f’(x_0)]^2